Newspace parameters
| Level: | \( N \) | \(=\) | \( 312 = 2^{3} \cdot 3 \cdot 13 \) |
| Weight: | \( k \) | \(=\) | \( 2 \) |
| Character orbit: | \([\chi]\) | \(=\) | 312.g (of order \(2\), degree \(1\), minimal) |
Newform invariants
| Self dual: | no |
| Analytic conductor: | \(2.49133254306\) |
| Analytic rank: | \(0\) |
| Dimension: | \(8\) |
| Coefficient field: | \(\Q(\zeta_{20})\) |
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|
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| Defining polynomial: |
\( x^{8} - x^{6} + x^{4} - x^{2} + 1 \)
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| Coefficient ring: | \(\Z[a_1, \ldots, a_{5}]\) |
| Coefficient ring index: | \( 2^{4} \) |
| Twist minimal: | yes |
| Sato-Tate group: | $\mathrm{SU}(2)[C_{2}]$ |
$q$-expansion
Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{7}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.
Basis of coefficient ring
| \(\beta_{1}\) | \(=\) |
\( 2\zeta_{20} \)
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| \(\beta_{2}\) | \(=\) |
\( 2\zeta_{20}^{4} \)
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| \(\beta_{3}\) | \(=\) |
\( \zeta_{20}^{5} \)
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| \(\beta_{4}\) | \(=\) |
\( \zeta_{20}^{7} + \zeta_{20}^{2} \)
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| \(\beta_{5}\) | \(=\) |
\( -\zeta_{20}^{7} + \zeta_{20}^{2} \)
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| \(\beta_{6}\) | \(=\) |
\( \zeta_{20}^{6} - \zeta_{20}^{4} + \zeta_{20}^{3} + \zeta_{20}^{2} - 1 \)
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| \(\beta_{7}\) | \(=\) |
\( -\zeta_{20}^{6} + \zeta_{20}^{4} + \zeta_{20}^{3} - \zeta_{20}^{2} + 1 \)
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| \(\zeta_{20}\) | \(=\) |
\( ( \beta_1 ) / 2 \)
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| \(\zeta_{20}^{2}\) | \(=\) |
\( ( \beta_{5} + \beta_{4} ) / 2 \)
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| \(\zeta_{20}^{3}\) | \(=\) |
\( ( \beta_{7} + \beta_{6} ) / 2 \)
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| \(\zeta_{20}^{4}\) | \(=\) |
\( ( \beta_{2} ) / 2 \)
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| \(\zeta_{20}^{5}\) | \(=\) |
\( \beta_{3} \)
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| \(\zeta_{20}^{6}\) | \(=\) |
\( ( -\beta_{7} + \beta_{6} - \beta_{5} - \beta_{4} + \beta_{2} + 2 ) / 2 \)
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| \(\zeta_{20}^{7}\) | \(=\) |
\( ( -\beta_{5} + \beta_{4} ) / 2 \)
|
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/312\mathbb{Z}\right)^\times\).
| \(n\) | \(79\) | \(145\) | \(157\) | \(209\) |
| \(\chi(n)\) | \(1\) | \(1\) | \(-1\) | \(1\) |
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
| Label | \(\iota_m(\nu)\) | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||||||||||||||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 157.1 |
|
−1.26007 | − | 0.642040i | 1.00000i | 1.17557 | + | 1.61803i | 2.79360i | 0.642040 | − | 1.26007i | 1.28408 | −0.442463 | − | 2.79360i | −1.00000 | 1.79360 | − | 3.52015i | ||||||||||||||||||||||||||||||||
| 157.2 | −1.26007 | + | 0.642040i | − | 1.00000i | 1.17557 | − | 1.61803i | − | 2.79360i | 0.642040 | + | 1.26007i | 1.28408 | −0.442463 | + | 2.79360i | −1.00000 | 1.79360 | + | 3.52015i | |||||||||||||||||||||||||||||||
| 157.3 | 0.221232 | − | 1.39680i | 1.00000i | −1.90211 | − | 0.618034i | − | 2.52015i | 1.39680 | + | 0.221232i | 2.79360 | −1.28408 | + | 2.52015i | −1.00000 | −3.52015 | − | 0.557537i | ||||||||||||||||||||||||||||||||
| 157.4 | 0.221232 | + | 1.39680i | − | 1.00000i | −1.90211 | + | 0.618034i | 2.52015i | 1.39680 | − | 0.221232i | 2.79360 | −1.28408 | − | 2.52015i | −1.00000 | −3.52015 | + | 0.557537i | ||||||||||||||||||||||||||||||||
| 157.5 | 0.642040 | − | 1.26007i | − | 1.00000i | −1.17557 | − | 1.61803i | − | 0.442463i | −1.26007 | − | 0.642040i | −2.52015 | −2.79360 | + | 0.442463i | −1.00000 | −0.557537 | − | 0.284079i | |||||||||||||||||||||||||||||||
| 157.6 | 0.642040 | + | 1.26007i | 1.00000i | −1.17557 | + | 1.61803i | 0.442463i | −1.26007 | + | 0.642040i | −2.52015 | −2.79360 | − | 0.442463i | −1.00000 | −0.557537 | + | 0.284079i | |||||||||||||||||||||||||||||||||
| 157.7 | 1.39680 | − | 0.221232i | 1.00000i | 1.90211 | − | 0.618034i | 1.28408i | 0.221232 | + | 1.39680i | 0.442463 | 2.52015 | − | 1.28408i | −1.00000 | 0.284079 | + | 1.79360i | |||||||||||||||||||||||||||||||||
| 157.8 | 1.39680 | + | 0.221232i | − | 1.00000i | 1.90211 | + | 0.618034i | − | 1.28408i | 0.221232 | − | 1.39680i | 0.442463 | 2.52015 | + | 1.28408i | −1.00000 | 0.284079 | − | 1.79360i | |||||||||||||||||||||||||||||||
Inner twists
| Char | Parity | Ord | Mult | Type |
|---|---|---|---|---|
| 1.a | even | 1 | 1 | trivial |
| 8.b | even | 2 | 1 | inner |
Twists
| By twisting character orbit | |||||||
|---|---|---|---|---|---|---|---|
| Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
| 1.a | even | 1 | 1 | trivial | 312.2.g.a | ✓ | 8 |
| 3.b | odd | 2 | 1 | 936.2.g.d | 8 | ||
| 4.b | odd | 2 | 1 | 1248.2.g.a | 8 | ||
| 8.b | even | 2 | 1 | inner | 312.2.g.a | ✓ | 8 |
| 8.d | odd | 2 | 1 | 1248.2.g.a | 8 | ||
| 12.b | even | 2 | 1 | 3744.2.g.d | 8 | ||
| 16.e | even | 4 | 1 | 9984.2.a.y | 4 | ||
| 16.e | even | 4 | 1 | 9984.2.a.bb | 4 | ||
| 16.f | odd | 4 | 1 | 9984.2.a.s | 4 | ||
| 16.f | odd | 4 | 1 | 9984.2.a.bh | 4 | ||
| 24.f | even | 2 | 1 | 3744.2.g.d | 8 | ||
| 24.h | odd | 2 | 1 | 936.2.g.d | 8 | ||
| By twisted newform orbit | |||||||
|---|---|---|---|---|---|---|---|
| Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
| 312.2.g.a | ✓ | 8 | 1.a | even | 1 | 1 | trivial |
| 312.2.g.a | ✓ | 8 | 8.b | even | 2 | 1 | inner |
| 936.2.g.d | 8 | 3.b | odd | 2 | 1 | ||
| 936.2.g.d | 8 | 24.h | odd | 2 | 1 | ||
| 1248.2.g.a | 8 | 4.b | odd | 2 | 1 | ||
| 1248.2.g.a | 8 | 8.d | odd | 2 | 1 | ||
| 3744.2.g.d | 8 | 12.b | even | 2 | 1 | ||
| 3744.2.g.d | 8 | 24.f | even | 2 | 1 | ||
| 9984.2.a.s | 4 | 16.f | odd | 4 | 1 | ||
| 9984.2.a.y | 4 | 16.e | even | 4 | 1 | ||
| 9984.2.a.bb | 4 | 16.e | even | 4 | 1 | ||
| 9984.2.a.bh | 4 | 16.f | odd | 4 | 1 | ||
Hecke kernels
This newform subspace can be constructed as the kernel of the linear operator
\( T_{5}^{8} + 16T_{5}^{6} + 76T_{5}^{4} + 96T_{5}^{2} + 16 \)
acting on \(S_{2}^{\mathrm{new}}(312, [\chi])\).
Hecke characteristic polynomials
| $p$ | $F_p(T)$ |
|---|---|
| $2$ |
\( T^{8} - 2 T^{7} + \cdots + 16 \)
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| $3$ |
\( (T^{2} + 1)^{4} \)
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| $5$ |
\( T^{8} + 16 T^{6} + \cdots + 16 \)
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| $7$ |
\( (T^{4} - 2 T^{3} - 6 T^{2} + \cdots - 4)^{2} \)
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| $11$ |
\( T^{8} + 40 T^{6} + \cdots + 400 \)
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| $13$ |
\( (T^{2} + 1)^{4} \)
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| $17$ |
\( (T^{4} + 8 T^{3} + 4 T^{2} + \cdots + 16)^{2} \)
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| $19$ |
\( T^{8} + 64 T^{6} + \cdots + 16 \)
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| $23$ |
\( (T^{4} + 4 T^{3} - 24 T^{2} + \cdots + 16)^{2} \)
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| $29$ |
\( T^{8} + 104 T^{6} + \cdots + 256 \)
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| $31$ |
\( (T^{4} + 2 T^{3} + \cdots + 316)^{2} \)
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| $37$ |
\( T^{8} + 176 T^{6} + \cdots + 952576 \)
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| $41$ |
\( (T^{4} - 18 T^{3} + \cdots - 2644)^{2} \)
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| $43$ |
\( T^{8} + 344 T^{6} + \cdots + 49336576 \)
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| $47$ |
\( (T^{4} - 12 T^{3} + \cdots - 964)^{2} \)
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| $53$ |
\( T^{8} + 224 T^{6} + \cdots + 1478656 \)
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| $59$ |
\( T^{8} + 104 T^{6} + \cdots + 5776 \)
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| $61$ |
\( (T^{4} + 60 T^{2} + 400)^{2} \)
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| $67$ |
\( T^{8} + 176 T^{6} + \cdots + 5776 \)
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| $71$ |
\( (T^{4} - 8 T^{3} + \cdots + 956)^{2} \)
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| $73$ |
\( (T^{4} - 16 T^{3} + \cdots - 304)^{2} \)
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| $79$ |
\( (T^{4} - 120 T^{2} + \cdots + 80)^{2} \)
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| $83$ |
\( T^{8} + 184 T^{6} + \cdots + 2572816 \)
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| $89$ |
\( (T^{4} - 30 T^{3} + \cdots - 2420)^{2} \)
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| $97$ |
\( (T^{4} + 28 T^{3} + \cdots - 18544)^{2} \)
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